Equipotential surfaces
Surfaces of equal potential
An equipotential surface is a surface on which the potential has the same value at every point.
Around a single point charge, V = kq/r depends only on r. So the equipotential surfaces are spheres centred on the charge. In a flat drawing they appear as circles.
Draw them at equal steps of potential: 100 V, 200 V, 300 V and so on. They crowd together near the charge, where the potential changes fastest.
Now add the field lines. They point radially outward, and they cross every circle at a right angle. This is always true.
Here is why. Move a charge along an equipotential surface. Its potential does not change, so no work is done on it. That is possible only if the field has no component along the surface.
So at every point, the field is normal to the equipotential surface through that point.
In a uniform field along the x-axis, the equipotentials are planes perpendicular to x. For equal steps of potential they are equally spaced.
For a dipole, and for two equal positive charges, the equipotentials have more interesting shapes. But everywhere, the field lines still cut them at right angles.
Two different equipotential surfaces can never cross. At a crossing point the potential would have two values, and the field would need two directions at once.
Field from the potential
Take two equipotential surfaces very close together: one at potential V, the other at V + δV. The perpendicular distance between them is δl.
The field E points from the higher surface to the lower one, perpendicular to both. Carry a unit positive charge across, from the lower surface to the higher one, against the field.
The work needed is force times distance, |E| δl. It must also equal the rise in potential, δV. So |E| δl = δV.
Since the field points the way V decreases, we write E = −δV/δl. The size of E is how fast the potential changes per metre, measured perpendicular to the surfaces.
Two things follow. First, the electric field points in the direction in which the potential decreases most steeply.
Second, where the equipotentials are close together, the field is strong; where they are far apart, it is weak. A contour map shows you the field at a glance.
Worked example: in a uniform field, equipotentials 10 V apart are 2 mm apart. Then E = 10 ÷ 0.002, which is 5000 volts per metre, pointing towards lower potential.
This gives another unit for the field: the volt per metre, which is the same as the newton per coulomb.
Turn it round: if E is 2000 V/m, the potential drops by 2000 volts for every metre you move along the field. That is 20 volts per centimetre.
Try it: map the equipotentials
- The orange lines are equipotentials 100 V apart. The white field lines cross them at right angles everywhere.
- Press “Set A at q₀”, then drag q₀ along one equipotential line. The work stays close to zero.
- Where the lines crowd together, the field arrow on q₀ is long; where they spread out, it is short.
- Halfway between the two charges, E is zero but V is not. Drag q₀ there and check.
Read the spacing
In a uniform field, equipotential surfaces 5 V apart are 0.5 cm apart. What is the electric field?
- 1000 V/m, towards lower potential
- 10 V/m, towards lower potential
- 1000 V/m, towards higher potential
- 2.5 V/m, along the equipotentials
Answer
A. |E| = δV/δl = 5 V ÷ 0.005 m = 1000 V/m. Remember to change cm to m (10 V/m forgets this). E is perpendicular to the equipotentials and points the way the potential decreases: E = −δV/δl.
Can they cross?
Why can two different equipotential surfaces never intersect?
- Because field lines would have to end there
- Because at the crossing the potential would have two values and E two directions
- Because equipotential surfaces are always spheres
- They can intersect, but only inside a conductor
Answer
B. The potential at a point has one value. At a crossing it would have to equal both V₁ and V₂. Also E must be normal to each surface, so it would need two directions at once. Equipotentials are spheres only around a single point charge.
Summary
- Field from potential:
E = −δV/δl - Size of the field:
|E| = |δV|/δl - Units:
1 V/m = 1 N/C
- An equipotential surface has the same potential at every point; no work is done in moving a charge along it.
- The field is always normal to the equipotential surfaces and points towards decreasing potential.
- Point charge: concentric spheres. Uniform field: parallel planes perpendicular to E.
- Equipotentials drawn at equal steps are closer where the field is stronger, and never intersect.